© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a250 K11a252

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8493 X12,5,13,6 X2837 X16,10,17,9 X18,11,19,12 X4,13,5,14 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,21,11,22

Gauss Code:

{1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, -8, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 8 12 2 16 18 4 20 22 14 10

Alexander Polynomial:

t-4 - 6t-3 + 16t-2 - 27t-1 + 33 - 27t + 16t2 - 6t3 + t4

Conway Polynomial:

1 - z2 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a253, ...}

Determinant and Signature:

{133, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 13q-3 + 18q-2 - 21q-1 + 22 - 19q + 15q2 - 9q3 + 4q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a228, K11a253, ...}

A2 (sl(3)) Invariant:

q-18 + 2q-12 - 4q-10 + 2q-8 - 2q-6 - 2q-4 + 4q-2 - 3 + 6q2 - 2q4 + q6 + 2q8 - 3q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - 3a-2z4 - a-2z6 + 5 + 10z2 + 10z4 + 5z6 + z8 - 5a2 - 11a2z2 - 8a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 - a-3z + 5a-3z3 - 12a-3z5 + 8a-3z7 + a-2 - 6a-2z2 + 12a-2z4 - 17a-2z6 + 10a-2z8 - 2a-1z + 7a-1z3 - 6a-1z5 - 5a-1z7 + 7a-1z9 + 5 - 23z2 + 46z4 - 42z6 + 14z8 + 2z10 - 2az - az3 + 16az5 - 24az7 + 12az9 + 5a2 - 24a2z2 + 42a2z4 - 34a2z6 + 9a2z8 + 2a2z10 - 3a3z + 4a3z3 + a3z5 - 8a3z7 + 5a3z9 + 2a4 - 6a4z2 + 10a4z4 - 12a4z6 + 5a4z8 - 2a5z + 6a5z3 - 8a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11251. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   6 1   j = 5                 9 3     j = 3               10 6       j = 1             12 9         j = -1           10 11           j = -3         8 11             j = -5       5 10               j = -7     2 8                 j = -9   1 5                   j = -11   2                     j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 251]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 251]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[4, 13, 5, 14], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[14, 19, 15, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 251]]
Out[4]=	GaussCode[1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 251]]
Out[5]=	DTCode[6, 8, 12, 2, 16, 18, 4, 20, 22, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 251]][t]
Out[6]=	-4 6 16 27 2 3 4 33 + t - -- + -- - -- - 27 t + 16 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 251]][z]
Out[7]=	2 6 8 1 - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 251], Knot[11, Alternating, 253]}
In[9]:=	{KnotDet[Knot[11, Alternating, 251]], KnotSignature[Knot[11, Alternating, 251]]}
Out[9]=	{133, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 251]][q]
Out[10]=	-6 3 7 13 18 21 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 19 q + 15 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 228], Knot[11, Alternating, 251], > Knot[11, Alternating, 253]}
In[12]:=	A2Invariant[Knot[11, Alternating, 251]][q]
Out[12]=	-18 2 4 2 2 2 4 2 4 6 8 10 -3 + q + --- - --- + -- - -- - -- + -- + 6 q - 2 q + q + 2 q - 3 q + 12 10 8 6 4 2 q q q q q q 12 14 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 251]][a, z]
Out[13]=	2 4 -2 2 4 2 3 z 2 2 4 2 4 3 z 5 - a - 5 a + 2 a + 10 z - ---- - 11 a z + 3 a z + 10 z - ---- - 2 2 a a 6 2 4 4 4 6 z 2 6 8 > 8 a z + a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 251]][a, z]
Out[14]=	2 -2 2 4 z 2 z 3 5 2 z 5 + a + 5 a + 2 a - -- - --- - 2 a z - 3 a z - 2 a z - 23 z + -- - 3 a 4 a a 2 3 3 3 6 z 2 2 4 2 6 2 z 5 z 7 z 3 3 3 > ---- - 24 a z - 6 a z + 2 a z - -- + ---- + ---- - a z + 4 a z + 2 5 3 a a a a 4 4 5 5 3 4 5 z 12 z 2 4 4 4 6 4 z > 6 a z + 46 z - ---- + ----- + 42 a z + 10 a z - 3 a z + -- - 4 2 5 a a a 5 5 6 6 12 z 6 z 5 3 5 5 5 6 4 z 17 z > ----- - ---- + 16 a z + a z - 8 a z - 42 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 8 z 5 z 7 3 7 5 7 > 34 a z - 12 a z + a z + ---- - ---- - 24 a z - 8 a z + 3 a z + 3 a a 8 9 8 10 z 2 8 4 8 7 z 9 3 9 10 > 14 z + ----- + 9 a z + 5 a z + ---- + 12 a z + 5 a z + 2 z + 2 a a 2 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 251]], Vassiliev[3][Knot[11, Alternating, 251]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 251]][q, t]
Out[16]=	11 1 2 1 5 2 8 5 10 -- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 8 11 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 10 q t + 6 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a251

K11a250 K11a252